On Monday, 16 House Republicans, led by Indiana Rep. Jackie Walorski, wrote to Obama asking him to remove Shinseki and two others in the VA's top leadership.

"Recent revelations about avoidable patient deaths, delayed treatments, falsified records, secret waiting lists, and cancelled appointments, coupled with systemic IT security failures, project cost overruns, and a backlog that has more than doubled since 2009, clearly demonstrate widespread incompetence and a lack of transparency within the Department of Veterans Affairs. ... Enough is enough," the lawmakers wrote. However, Boehner reiterated Monday that he does not believe Shinseki should be forced out of his position.

"I think it’d be the easiest thing in the world for the administration to do, take Shinseki out, go through a process of coming up with a new secretary, when that’s not the issue. There is a systemic problem within the organization in the VA," Boehner said at an event at the San Antonio Hispanic Chamber of Commerce.

Instead, Boehner said he would like to see Shinseki given more power to fire ineffective managers and cut down on bonuses.

"The problem is much, much bigger than who the secretary is," he said. "It will distract everyone’s attention if Shinseki goes and we wait around for a new secretary, and I don’t want that to happen. I want to keep the focus on fixing the problem, not fixing the personality."

"The president takes the situation, as he has said, around the Phoenix office very seriously," he said. "And that's why he directed Secretary Shinseki to investigate. And Secretary Shinseki has invited the independent Veterans Affairs inspector general — Office of Inspector General, to conduct a comprehensive review.

"The president remains confident that Secretary Shinseki is focused on this matter, and he's confident in Secretary Shinseki's ability to lead the department and take appropriate action based on the I.G.'s findings."

Carney noted that the backlog grew in part due to allowing Agent Orange claims and post-traumatic stress disorder claims for affected veterans.

Lawmakers in both parties have not been satisfied, although Republicans have been leading the charge for resignation, including the most recent letter, available here.Daniel Newhauser contributed to this report.